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Simplifying 84x2 + -238x + -196 = x Reorder the terms: -196 + -238x + 84x2 = x Solving -196 + -238x + 84x2 = x Solving for variable 'x'. Reorder the terms: -196 + -238x + -1x + 84x2 = x + -1x Combine like terms: -238x + -1x = -239x -196 + -239x + 84x2 = x + -1x Combine like terms: x + -1x = 0 -196 + -239x + 84x2 = 0 Begin completing the square. Divide all terms by 84 the coefficient of the squared term: Divide each side by '84'. -2.333333333 + -2.845238095x + x2 = 0 Move the constant term to the right: Add '2.333333333' to each side of the equation. -2.333333333 + -2.845238095x + 2.333333333 + x2 = 0 + 2.333333333 Reorder the terms: -2.333333333 + 2.333333333 + -2.845238095x + x2 = 0 + 2.333333333 Combine like terms: -2.333333333 + 2.333333333 = 0.000000000 0.000000000 + -2.845238095x + x2 = 0 + 2.333333333 -2.845238095x + x2 = 0 + 2.333333333 Combine like terms: 0 + 2.333333333 = 2.333333333 -2.845238095x + x2 = 2.333333333 The x term is -2.845238095x. Take half its coefficient (-1.422619048). Square it (2.023844956) and add it to both sides. Add '2.023844956' to each side of the equation. -2.845238095x + 2.023844956 + x2 = 2.333333333 + 2.023844956 Reorder the terms: 2.023844956 + -2.845238095x + x2 = 2.333333333 + 2.023844956 Combine like terms: 2.333333333 + 2.023844956 = 4.357178289 2.023844956 + -2.845238095x + x2 = 4.357178289 Factor a perfect square on the left side: (x + -1.422619048)(x + -1.422619048) = 4.357178289 Calculate the square root of the right side: 2.087385515 Break this problem into two subproblems by setting (x + -1.422619048) equal to 2.087385515 and -2.087385515.Subproblem 1
x + -1.422619048 = 2.087385515 Simplifying x + -1.422619048 = 2.087385515 Reorder the terms: -1.422619048 + x = 2.087385515 Solving -1.422619048 + x = 2.087385515 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.422619048' to each side of the equation. -1.422619048 + 1.422619048 + x = 2.087385515 + 1.422619048 Combine like terms: -1.422619048 + 1.422619048 = 0.000000000 0.000000000 + x = 2.087385515 + 1.422619048 x = 2.087385515 + 1.422619048 Combine like terms: 2.087385515 + 1.422619048 = 3.510004563 x = 3.510004563 Simplifying x = 3.510004563Subproblem 2
x + -1.422619048 = -2.087385515 Simplifying x + -1.422619048 = -2.087385515 Reorder the terms: -1.422619048 + x = -2.087385515 Solving -1.422619048 + x = -2.087385515 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.422619048' to each side of the equation. -1.422619048 + 1.422619048 + x = -2.087385515 + 1.422619048 Combine like terms: -1.422619048 + 1.422619048 = 0.000000000 0.000000000 + x = -2.087385515 + 1.422619048 x = -2.087385515 + 1.422619048 Combine like terms: -2.087385515 + 1.422619048 = -0.664766467 x = -0.664766467 Simplifying x = -0.664766467Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.510004563, -0.664766467}
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